403:, is known, it becomes easier to handle different pipe-flow problems, viz. calculating the pressure drop for evaluating pumping costs or to find the flow-rate in a piping network for a given pressure drop. It is usually extremely difficult to arrive at exact analytical solution to calculate the friction factor associated with flow of non-Newtonian fluids and therefore explicit approximations are used to calculate it. Once the friction factor has been calculated the pressure drop can be easily determined for a given flow by the
128:
101:
20:
1664:
120:, is reached. Beyond this point the flow rate increases steadily with increasing shear stress. This is roughly the way in which Bingham presented his observation, in an experimental study of paints. These properties allow a Bingham plastic to have a textured surface with peaks and ridges instead of a featureless surface like a
389:
1996:
Both Swamee–Aggarwal equation and the Darby–Melson equation can be combined to give an explicit equation for determining the friction factor of
Bingham plastic fluids in any regime. Relative roughness is not a parameter in any of the equations because the friction factor of Bingham plastic fluids is
146:
on the horizontal one. (Volumetric flow rate depends on the size of the pipe, shear rate is a measure of how the velocity changes with distance. It is proportional to flow rate, but does not depend on pipe size.) As before, the
Newtonian fluid flows and gives a shear rate for any finite value of
1368:
1392:
258:
1758:
111:
shows a graph of the behaviour of an ordinary viscous (or
Newtonian) fluid in red, for example in a pipe. If the pressure at one end of a pipe is increased this produces a stress on the fluid tending to make it move (called the
170:, and a certain amount of stress is required to break this structure. Once the structure has been broken, the particles move with the liquid under viscous forces. If the stress is removed, the particles associate again.
966:
1114:
1844:
766:
1270:
1659:{\displaystyle f_{L}={\frac {K_{1}+{\dfrac {4K_{2}}{\left(K_{1}+{\frac {K_{1}K_{2}}{K_{1}^{4}+3K_{2}}}\right)^{3}}}}{1+{\dfrac {3K_{2}}{\left(K_{1}+{\frac {K_{1}K_{2}}{K_{1}^{4}+3K_{2}}}\right)^{4}}}}}}
147:
shear stress. However, the
Bingham plastic again does not exhibit any shear rate (no flow and thus no velocity) until a certain stress is achieved. For the Newtonian fluid the slope of this line is the
902:
1991:
471:
1946:
1859:
In 1981, Darby and Melson, using the approach of
Churchill and of Churchill and Usagi, developed an expression to get a single friction factor equation valid for all flow regimes:
847:
821:
1236:
Note: Darby and Melson's expression is for a
Fanning friction factor, and needs to be multiplied by 4 to be used in the friction loss equations located elsewhere on this page.
116:) and the volumetric flow rate increases proportionally. However, for a Bingham Plastic fluid (in blue), stress can be applied but it will not flow until a certain value, the
1231:
1147:
799:
530:
1043:
384:{\displaystyle {\frac {\partial u}{\partial y}}={\begin{cases}0,&\tau <\tau _{0}\\{\frac {\tau -\tau _{0}}{\mu _{\infty }}},&\tau \geq \tau _{0}\end{cases}}}
226:
1248:, due to complexity of the solution it is rarely employed. Therefore, researchers have tried to develop explicit approximations for the Buckingham–Reiner equation.
992:
199:
1014:
1675:
622:
600:
578:
556:
497:
2238:(1981). "Approximate explicit analytical expressions of friction factor for flow of Bingham fluids in smooth pipes using Adomian decomposition method".
633:
An exact description of friction loss for
Bingham plastics in fully developed laminar pipe flow was first published by Buckingham. His expression, the
909:
1060:
1769:
1244:
Although an exact analytical solution of the
Buckingham–Reiner equation can be obtained because it is a fourth order polynomial equation in
1264:
equation, but the discrepancy from experimental data is well within the accuracy of the data. The Swamee–Aggarwal equation is given by:
643:
1363:{\displaystyle f_{L}={64 \over \mathrm {Re} }+{64 \over \mathrm {Re} }\left({\mathrm {He} \over 6.2218\mathrm {Re} }\right)^{0.958}}
399:
In fluid flow, it is a common problem to calculate the pressure drop in an established piping network. Once the friction factor,
863:
1957:
2270:
Churchill, S.W.; Usagi, R.A. (1972). "A general expression for the correlation of rates of transfer and other phenomena".
151:, which is the only parameter needed to describe its flow. By contrast, the Bingham plastic requires two parameters, the
162:
The physical reason for this behaviour is that the liquid contains particles (such as clay) or large molecules (such as
413:
2145:
2095:
1386:
by using the
Adomian decomposition method. The friction factor containing two terms through this method is given as:
2326:
2316:
1865:
2016:
832:
806:
404:
26:
is a
Bingham plastic. The surface has ridges and peaks because Bingham plastics mimic solids under low
2213:
Swamee, P.K. and Aggarwal, N.(2011). "Explicit equations for laminar flow of Bingham plastic fluids".
252:), is directly proportional to the amount by which the applied shear stress exceeds the yield stress:
290:
2011:
2182:
Darby, R. and Melson J.(1981). "How to predict the friction factor for flow of Bingham plastics".
1154:
1125:
777:
508:
1021:
204:
1256:
The Swamee–Aggarwal equation is used to solve directly for the Darcy–Weisbach friction factor
2255:
Churchill, S.W. (November 7, 1977). "Friction factor equation spans all fluid-flow regimes".
500:
166:) which have some kind of interaction, creating a weak solid structure, formerly known as a
2279:
1054:
Darby and Melson developed an empirical expression that was then refined, and is given by:
977:
184:
77:
1753:{\displaystyle K_{1}={16 \over \mathrm {Re} }+{16\mathrm {He} \over 6\mathrm {Re} ^{2}},}
999:
8:
233:
2283:
2321:
2311:
2112:
607:
585:
563:
541:
482:
69:
51:
2199:
Darby, R.; et al. (September 1992). "Prediction friction loss in slurry pipes".
2141:
2091:
62:
35:
2287:
2218:
2068:
2063:
2055:
1260:
for laminar flow of Bingham plastic fluids. It is an approximation of the implicit
2222:
854:
824:
121:
2026:
2006:
2305:
2137:
1849:
92:
is applied to the tube. It is then pushed out as a relatively coherent plug.
229:
179:
139:
138:
shows the way in which it is normally presented currently. The graph shows
117:
113:
73:
43:
27:
2291:
2118:
2059:
961:{\displaystyle \operatorname {He} ={\rho D^{2}\tau _{o} \over \mu ^{2}}}
16:
Material which is solid at low stress but becomes viscous at high stress
1109:{\displaystyle f_{\text{T}}=4\times 10^{a}\operatorname {Re} ^{-0.193}}
237:
143:
85:
47:
23:
2046:
Bingham, E.C. (1916). "An Investigation of the Laws of Plastic Flow".
1839:{\displaystyle K_{2}=-{16\mathrm {He} ^{4} \over 3\mathrm {Re} ^{8}}.}
1382:
have provided an explicit procedure to calculate the friction factor
249:
148:
55:
2021:
1239:
801:
is the laminar flow Darcy friction factor (SI units: dimensionless)
533:
89:
2161:
Buckingham, E. (1921). "On Plastic Flow Through Capillary Tubes".
163:
761:{\displaystyle f_{\text{L}}={64 \over \operatorname {Re} }\left}
1149:
is the turbulent flow friction factor (SI units: dimensionless)
81:
19:
637:
equation, can be written in a dimensionless form as follows:
58:
2240:
Communications in Nonlinear Science and Numerical Simulation
127:
100:
1045:
is the yield point (yield strength) of fluid (SI units: Pa)
377:
1850:
Combined equation for friction factor for all flow regimes
236:") is exceeded, the material flows in such a way that the
897:{\displaystyle \operatorname {Re} ={\rho VD \over \mu },}
1986:{\displaystyle m=1.7+{40000 \over \operatorname {Re} }}
104:
Figure 1. Bingham Plastic flow as described by Bingham
1960:
1868:
1772:
1678:
1544:
1426:
1395:
1273:
1157:
1128:
1063:
1024:
1002:
980:
912:
866:
857:
and the Hedstrom number are respectively defined as:
835:
809:
780:
646:
610:
588:
566:
544:
511:
485:
416:
261:
207:
187:
131:
Figure 2. Bingham Plastic flow as described currently
1016:
is the dynamic viscosity of fluid (SI units: kg/m s)
1985:
1940:
1838:
1752:
1658:
1362:
1225:
1141:
1108:
1037:
1008:
986:
960:
896:
841:
815:
793:
760:
616:
594:
572:
558:is the gravitational acceleration (SI units: m/s²)
550:
524:
491:
465:
383:
220:
193:
2303:
1240:Approximations of the Buckingham–Reiner equation
849:is the Hedstrom number (SI units: dimensionless)
466:{\displaystyle f={2h_{\text{f}}gD \over LV^{2}}}
2088:Rheological Methods in Food Process Engineering
2269:
994:is the mass density of fluid (SI units: kg/m)
88:, which will not be extruded until a certain
2215:Journal of Petroleum Science and Engineering
1251:
394:
2160:
624:is the mean fluid velocity (SI units: m/s)
2254:
2067:
2192:
2081:
2079:
1854:
1373:
155:and the slope of the line, known as the
126:
99:
18:
2178:
2176:
2110:
2045:
2304:
2085:
1941:{\displaystyle f=\left^{\frac {1}{m}}}
2198:
2131:
2076:
178:The material is an elastic solid for
2173:
2134:Chemical Engineering Fluid Mechanics
65:who proposed its mathematical form.
2048:Bulletin of the Bureau of Standards
842:{\displaystyle \operatorname {He} }
816:{\displaystyle \operatorname {Re} }
13:
1820:
1817:
1800:
1797:
1734:
1731:
1720:
1717:
1701:
1698:
1343:
1340:
1331:
1328:
1314:
1311:
1296:
1293:
580:is the pipe diameter (SI units: m)
346:
273:
265:
61:at high stress. It is named after
14:
2338:
1997:not sensitive to pipe roughness.
1049:
2132:Darby, Ron (1996). "Chapter 6".
602:is the pipe length (SI units: m)
2263:
2248:
2090:(2nd ed.). Freeman Press.
628:
2228:
2207:
2154:
2125:
2104:
2039:
95:
1:
2032:
532:is the frictional head loss (
201:, less than a critical value
173:
2223:10.1016/j.petrol.2011.01.015
1226:{\displaystyle a=-1.47\left}
1142:{\displaystyle f_{\text{T}}}
794:{\displaystyle f_{\text{L}}}
525:{\displaystyle h_{\text{f}}}
7:
2017:Bingham-Papanastasiou model
2000:
46:material that behaves as a
10:
2343:
1038:{\displaystyle \tau _{o}}
827:(SI units: dimensionless)
503:(SI units: dimensionless)
221:{\displaystyle \tau _{0}}
142:on the vertical axis and
80:, and in the handling of
1252:Swamee–Aggarwal equation
395:Friction factor formulae
2114:Fluidity and Plasticity
2069:2027/mdp.39015086559054
405:Darcy–Weisbach equation
68:It is used as a common
2111:Bingham, E.C. (1922).
1987:
1942:
1840:
1754:
1660:
1364:
1227:
1143:
1110:
1039:
1010:
988:
962:
898:
843:
817:
795:
762:
618:
596:
574:
552:
526:
493:
467:
385:
222:
195:
132:
105:
84:. A common example is
31:
2292:10.1002/aic.690180606
2086:Steffe, J.F. (1996).
2012:Bernoulli's principle
1988:
1943:
1855:Darby–Melson equation
1841:
1755:
1661:
1374:Danish–Kumar solution
1365:
1228:
1144:
1111:
1040:
1011:
989:
987:{\displaystyle \rho }
963:
899:
844:
818:
796:
763:
619:
597:
575:
553:
527:
501:Darcy friction factor
494:
468:
386:
223:
196:
194:{\displaystyle \tau }
130:
103:
22:
2327:Offshore engineering
2317:Non-Newtonian fluids
2257:Chemical Engineering
2201:Chemical Engineering
2184:Chemical Engineering
2060:10.6028/bulletin.304
1958:
1866:
1770:
1676:
1393:
1271:
1155:
1126:
1061:
1022:
1009:{\displaystyle \mu }
1000:
978:
910:
864:
833:
807:
778:
644:
608:
586:
564:
542:
509:
483:
414:
259:
250:article on viscosity
228:. Once the critical
205:
185:
78:drilling engineering
2284:1972AIChE..18.1121C
1619:
1501:
248:(as defined in the
1983:
1938:
1836:
1750:
1656:
1651:
1605:
1533:
1487:
1360:
1223:
1139:
1106:
1035:
1006:
984:
958:
894:
839:
813:
791:
758:
614:
592:
570:
548:
522:
489:
463:
381:
376:
218:
191:
133:
106:
70:mathematical model
32:
1981:
1935:
1912:
1890:
1831:
1745:
1705:
1654:
1650:
1637:
1532:
1519:
1348:
1318:
1300:
1262:Buckingham–Reiner
1136:
1071:
956:
889:
788:
747:
707:
694:
668:
654:
635:Buckingham–Reiner
617:{\displaystyle V}
595:{\displaystyle L}
573:{\displaystyle D}
551:{\displaystyle g}
519:
492:{\displaystyle f}
461:
436:
351:
280:
157:plastic viscosity
63:Eugene C. Bingham
36:materials science
2334:
2296:
2295:
2278:(6): 1121–1128.
2267:
2261:
2260:
2252:
2246:
2232:
2226:
2211:
2205:
2204:
2196:
2190:
2180:
2171:
2170:
2163:ASTM Proceedings
2158:
2152:
2151:
2129:
2123:
2122:
2108:
2102:
2101:
2083:
2074:
2073:
2071:
2043:
1992:
1990:
1989:
1984:
1982:
1974:
1947:
1945:
1944:
1939:
1937:
1936:
1928:
1926:
1922:
1921:
1920:
1915:
1914:
1913:
1910:
1899:
1898:
1893:
1892:
1891:
1888:
1845:
1843:
1842:
1837:
1832:
1830:
1829:
1828:
1823:
1810:
1809:
1808:
1803:
1790:
1782:
1781:
1759:
1757:
1756:
1751:
1746:
1744:
1743:
1742:
1737:
1724:
1723:
1711:
1706:
1704:
1693:
1688:
1687:
1665:
1663:
1662:
1657:
1655:
1653:
1652:
1649:
1648:
1643:
1639:
1638:
1636:
1635:
1634:
1618:
1613:
1603:
1602:
1601:
1592:
1591:
1581:
1576:
1575:
1560:
1559:
1558:
1545:
1535:
1534:
1531:
1530:
1525:
1521:
1520:
1518:
1517:
1516:
1500:
1495:
1485:
1484:
1483:
1474:
1473:
1463:
1458:
1457:
1442:
1441:
1440:
1427:
1421:
1420:
1410:
1405:
1404:
1369:
1367:
1366:
1361:
1359:
1358:
1353:
1349:
1347:
1346:
1334:
1326:
1319:
1317:
1306:
1301:
1299:
1288:
1283:
1282:
1232:
1230:
1229:
1224:
1222:
1218:
1217:
1216:
1212:
1211:
1210:
1148:
1146:
1145:
1140:
1138:
1137:
1134:
1115:
1113:
1112:
1107:
1105:
1104:
1092:
1091:
1073:
1072:
1069:
1044:
1042:
1041:
1036:
1034:
1033:
1015:
1013:
1012:
1007:
993:
991:
990:
985:
967:
965:
964:
959:
957:
955:
954:
945:
944:
943:
934:
933:
920:
903:
901:
900:
895:
890:
885:
874:
848:
846:
845:
840:
822:
820:
819:
814:
800:
798:
797:
792:
790:
789:
786:
767:
765:
764:
759:
757:
753:
752:
748:
746:
745:
744:
735:
734:
724:
723:
714:
708:
700:
695:
693:
682:
669:
661:
656:
655:
652:
623:
621:
620:
615:
601:
599:
598:
593:
579:
577:
576:
571:
557:
555:
554:
549:
531:
529:
528:
523:
521:
520:
517:
498:
496:
495:
490:
472:
470:
469:
464:
462:
460:
459:
458:
445:
438:
437:
434:
424:
390:
388:
387:
382:
380:
379:
373:
372:
352:
350:
349:
340:
339:
338:
322:
316:
315:
281:
279:
271:
263:
227:
225:
224:
219:
217:
216:
200:
198:
197:
192:
2342:
2341:
2337:
2336:
2335:
2333:
2332:
2331:
2302:
2301:
2300:
2299:
2268:
2264:
2253:
2249:
2233:
2229:
2212:
2208:
2197:
2193:
2181:
2174:
2159:
2155:
2148:
2130:
2126:
2109:
2105:
2098:
2084:
2077:
2044:
2040:
2035:
2003:
1973:
1959:
1956:
1955:
1927:
1916:
1909:
1905:
1904:
1903:
1894:
1887:
1883:
1882:
1881:
1880:
1876:
1875:
1867:
1864:
1863:
1857:
1852:
1824:
1816:
1815:
1811:
1804:
1796:
1795:
1791:
1789:
1777:
1773:
1771:
1768:
1767:
1738:
1730:
1729:
1725:
1716:
1712:
1710:
1697:
1692:
1683:
1679:
1677:
1674:
1673:
1644:
1630:
1626:
1614:
1609:
1604:
1597:
1593:
1587:
1583:
1582:
1580:
1571:
1567:
1566:
1562:
1561:
1554:
1550:
1546:
1543:
1536:
1526:
1512:
1508:
1496:
1491:
1486:
1479:
1475:
1469:
1465:
1464:
1462:
1453:
1449:
1448:
1444:
1443:
1436:
1432:
1428:
1425:
1416:
1412:
1411:
1409:
1400:
1396:
1394:
1391:
1390:
1376:
1354:
1339:
1335:
1327:
1325:
1321:
1320:
1310:
1305:
1292:
1287:
1278:
1274:
1272:
1269:
1268:
1254:
1242:
1203:
1199:
1198:
1188:
1184:
1174:
1170:
1156:
1153:
1152:
1133:
1129:
1127:
1124:
1123:
1097:
1093:
1087:
1083:
1068:
1064:
1062:
1059:
1058:
1052:
1029:
1025:
1023:
1020:
1019:
1001:
998:
997:
979:
976:
975:
950:
946:
939:
935:
929:
925:
921:
919:
911:
908:
907:
875:
873:
865:
862:
861:
855:Reynolds number
834:
831:
830:
825:Reynolds number
808:
805:
804:
785:
781:
779:
776:
775:
740:
736:
730:
726:
725:
719:
715:
713:
709:
699:
686:
681:
674:
670:
660:
651:
647:
645:
642:
641:
631:
609:
606:
605:
587:
584:
583:
565:
562:
561:
543:
540:
539:
516:
512:
510:
507:
506:
484:
481:
480:
454:
450:
446:
433:
429:
425:
423:
415:
412:
411:
397:
375:
374:
368:
364:
356:
345:
341:
334:
330:
323:
321:
318:
317:
311:
307:
299:
286:
285:
272:
264:
262:
260:
257:
256:
212:
208:
206:
203:
202:
186:
183:
182:
176:
122:Newtonian fluid
98:
54:but flows as a
40:Bingham plastic
17:
12:
11:
5:
2340:
2330:
2329:
2324:
2319:
2314:
2298:
2297:
2262:
2247:
2227:
2206:
2191:
2172:
2153:
2146:
2124:
2121:. p. 219.
2103:
2096:
2075:
2054:(2): 309–353.
2037:
2036:
2034:
2031:
2030:
2029:
2027:Shear thinning
2024:
2019:
2014:
2009:
2007:Bagnold number
2002:
1999:
1994:
1993:
1980:
1977:
1972:
1969:
1966:
1963:
1949:
1948:
1934:
1931:
1925:
1919:
1908:
1902:
1897:
1886:
1879:
1874:
1871:
1856:
1853:
1851:
1848:
1847:
1846:
1835:
1827:
1822:
1819:
1814:
1807:
1802:
1799:
1794:
1788:
1785:
1780:
1776:
1761:
1760:
1749:
1741:
1736:
1733:
1728:
1722:
1719:
1715:
1709:
1703:
1700:
1696:
1691:
1686:
1682:
1667:
1666:
1647:
1642:
1633:
1629:
1625:
1622:
1617:
1612:
1608:
1600:
1596:
1590:
1586:
1579:
1574:
1570:
1565:
1557:
1553:
1549:
1542:
1539:
1529:
1524:
1515:
1511:
1507:
1504:
1499:
1494:
1490:
1482:
1478:
1472:
1468:
1461:
1456:
1452:
1447:
1439:
1435:
1431:
1424:
1419:
1415:
1408:
1403:
1399:
1375:
1372:
1371:
1370:
1357:
1352:
1345:
1342:
1338:
1333:
1330:
1324:
1316:
1313:
1309:
1304:
1298:
1295:
1291:
1286:
1281:
1277:
1253:
1250:
1241:
1238:
1234:
1233:
1221:
1215:
1209:
1206:
1202:
1197:
1194:
1191:
1187:
1183:
1180:
1177:
1173:
1169:
1166:
1163:
1160:
1150:
1132:
1117:
1116:
1103:
1100:
1096:
1090:
1086:
1082:
1079:
1076:
1067:
1051:
1050:Turbulent flow
1048:
1047:
1046:
1032:
1028:
1017:
1005:
995:
983:
969:
968:
953:
949:
942:
938:
932:
928:
924:
918:
915:
905:
893:
888:
884:
881:
878:
872:
869:
851:
850:
838:
828:
812:
802:
784:
769:
768:
756:
751:
743:
739:
733:
729:
722:
718:
712:
706:
703:
698:
692:
689:
685:
680:
677:
673:
667:
664:
659:
650:
630:
627:
626:
625:
613:
603:
591:
581:
569:
559:
547:
537:
515:
504:
488:
474:
473:
457:
453:
449:
444:
441:
432:
428:
422:
419:
396:
393:
392:
391:
378:
371:
367:
363:
360:
357:
355:
348:
344:
337:
333:
329:
326:
320:
319:
314:
310:
306:
303:
300:
298:
295:
292:
291:
289:
284:
278:
275:
270:
267:
215:
211:
190:
175:
172:
97:
94:
28:shear stresses
15:
9:
6:
4:
3:
2:
2339:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2309:
2307:
2293:
2289:
2285:
2281:
2277:
2273:
2272:AIChE Journal
2266:
2258:
2251:
2244:
2241:
2237:
2231:
2224:
2220:
2216:
2210:
2202:
2195:
2188:
2185:
2179:
2177:
2168:
2164:
2157:
2149:
2147:0-8247-0444-4
2143:
2139:
2138:Marcel Dekker
2135:
2128:
2120:
2116:
2115:
2107:
2099:
2097:0-9632036-1-4
2093:
2089:
2082:
2080:
2070:
2065:
2061:
2057:
2053:
2049:
2042:
2038:
2028:
2025:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2004:
1998:
1978:
1975:
1970:
1967:
1964:
1961:
1954:
1953:
1952:
1932:
1929:
1923:
1917:
1906:
1900:
1895:
1884:
1877:
1872:
1869:
1862:
1861:
1860:
1833:
1825:
1812:
1805:
1792:
1786:
1783:
1778:
1774:
1766:
1765:
1764:
1747:
1739:
1726:
1713:
1707:
1694:
1689:
1684:
1680:
1672:
1671:
1670:
1645:
1640:
1631:
1627:
1623:
1620:
1615:
1610:
1606:
1598:
1594:
1588:
1584:
1577:
1572:
1568:
1563:
1555:
1551:
1547:
1540:
1537:
1527:
1522:
1513:
1509:
1505:
1502:
1497:
1492:
1488:
1480:
1476:
1470:
1466:
1459:
1454:
1450:
1445:
1437:
1433:
1429:
1422:
1417:
1413:
1406:
1401:
1397:
1389:
1388:
1387:
1385:
1381:
1355:
1350:
1336:
1322:
1307:
1302:
1289:
1284:
1279:
1275:
1267:
1266:
1265:
1263:
1259:
1249:
1247:
1237:
1219:
1213:
1207:
1204:
1200:
1195:
1192:
1189:
1185:
1181:
1178:
1175:
1171:
1167:
1164:
1161:
1158:
1151:
1130:
1122:
1121:
1120:
1101:
1098:
1094:
1088:
1084:
1080:
1077:
1074:
1065:
1057:
1056:
1055:
1030:
1026:
1018:
1003:
996:
981:
974:
973:
972:
951:
947:
940:
936:
930:
926:
922:
916:
913:
906:
891:
886:
882:
879:
876:
870:
867:
860:
859:
858:
856:
836:
829:
826:
810:
803:
782:
774:
773:
772:
754:
749:
741:
737:
731:
727:
720:
716:
710:
704:
701:
696:
690:
687:
683:
678:
675:
671:
665:
662:
657:
648:
640:
639:
638:
636:
611:
604:
589:
582:
567:
560:
545:
538:
535:
513:
505:
502:
486:
479:
478:
477:
455:
451:
447:
442:
439:
430:
426:
420:
417:
410:
409:
408:
406:
402:
369:
365:
361:
358:
353:
342:
335:
331:
327:
324:
312:
308:
304:
301:
296:
293:
287:
282:
276:
268:
255:
254:
253:
251:
247:
243:
239:
235:
231:
213:
209:
188:
181:
171:
169:
165:
160:
158:
154:
150:
145:
141:
137:
129:
125:
123:
119:
115:
110:
102:
93:
91:
87:
83:
79:
75:
71:
66:
64:
60:
57:
53:
49:
45:
41:
37:
29:
25:
21:
2275:
2271:
2265:
2256:
2250:
2242:
2239:
2235:
2230:
2214:
2209:
2200:
2194:
2186:
2183:
2169:: 1154–1156.
2166:
2162:
2156:
2133:
2127:
2117:. New York:
2113:
2106:
2087:
2051:
2047:
2041:
1995:
1950:
1858:
1762:
1668:
1383:
1379:
1377:
1261:
1257:
1255:
1245:
1243:
1235:
1118:
1053:
970:
852:
770:
634:
632:
629:Laminar flow
475:
400:
398:
245:
241:
234:yield stress
230:shear stress
180:shear stress
177:
167:
161:
156:
153:yield stress
152:
140:shear stress
135:
134:
118:yield stress
114:shear stress
108:
107:
67:
44:viscoplastic
39:
33:
2234:Danish, M.
2119:McGraw-Hill
96:Explanation
2306:Categories
2245:: 239–251.
2033:References
238:shear rate
174:Definition
168:false body
144:shear rate
86:toothpaste
48:rigid body
24:Mayonnaise
2322:Viscosity
2312:Materials
1787:−
1205:−
1196:×
1190:−
1165:−
1099:−
1081:×
1027:τ
1004:μ
982:ρ
948:μ
937:τ
923:ρ
887:μ
877:ρ
697:−
366:τ
362:≥
359:τ
347:∞
343:μ
332:τ
328:−
325:τ
309:τ
302:τ
274:∂
266:∂
210:τ
189:τ
149:viscosity
2259:: 91–92.
2189:: 59–61.
2022:Rheology
2001:See also
534:SI units
164:polymers
136:Figure 2
109:Figure 1
90:pressure
82:slurries
76:flow in
52:stresses
2280:Bibcode
1951:where:
1378:Danish
1119:where:
971:where:
823:is the
771:where:
499:is the
476:where:
56:viscous
50:at low
2236:et al.
2144:
2094:
1669:where
1380:et al.
1337:6.2218
1976:40000
1356:0.958
1182:0.146
1102:0.193
232:(or "
59:fluid
42:is a
2142:ISBN
2092:ISBN
1763:and
1168:1.47
853:The
536:: m)
305:<
38:, a
2288:doi
2219:doi
2064:hdl
2056:doi
1968:1.7
1193:2.9
904:and
240:, ∂
74:mud
72:of
34:In
2308::
2286:.
2276:18
2274:.
2243:16
2217:.
2187:28
2175:^
2167:21
2165:.
2140:.
2136:.
2078:^
2062:.
2052:13
2050:.
1979:Re
1793:16
1714:16
1695:16
1308:64
1290:64
1214:He
1201:10
1095:Re
1085:10
914:He
868:Re
837:He
811:Re
738:Re
717:He
702:64
691:Re
684:He
666:Re
663:64
407::
244:/∂
159:.
124:.
2294:.
2290::
2282::
2225:.
2221::
2203:.
2150:.
2100:.
2072:.
2066::
2058::
1971:+
1965:=
1962:m
1933:m
1930:1
1924:]
1918:m
1911:T
1907:f
1901:+
1896:m
1889:L
1885:f
1878:[
1873:=
1870:f
1834:.
1826:8
1821:e
1818:R
1813:3
1806:4
1801:e
1798:H
1784:=
1779:2
1775:K
1748:,
1740:2
1735:e
1732:R
1727:6
1721:e
1718:H
1708:+
1702:e
1699:R
1690:=
1685:1
1681:K
1646:4
1641:)
1632:2
1628:K
1624:3
1621:+
1616:4
1611:1
1607:K
1599:2
1595:K
1589:1
1585:K
1578:+
1573:1
1569:K
1564:(
1556:2
1552:K
1548:3
1541:+
1538:1
1528:3
1523:)
1514:2
1510:K
1506:3
1503:+
1498:4
1493:1
1489:K
1481:2
1477:K
1471:1
1467:K
1460:+
1455:1
1451:K
1446:(
1438:2
1434:K
1430:4
1423:+
1418:1
1414:K
1407:=
1402:L
1398:f
1384:f
1351:)
1344:e
1341:R
1332:e
1329:H
1323:(
1315:e
1312:R
1303:+
1297:e
1294:R
1285:=
1280:L
1276:f
1258:f
1246:f
1220:]
1208:5
1186:e
1179:+
1176:1
1172:[
1162:=
1159:a
1135:T
1131:f
1089:a
1078:4
1075:=
1070:T
1066:f
1031:o
952:2
941:o
931:2
927:D
917:=
892:,
883:D
880:V
871:=
787:L
783:f
755:]
750:)
742:7
732:3
728:f
721:4
711:(
705:3
688:6
679:+
676:1
672:[
658:=
653:L
649:f
612:V
590:L
568:D
546:g
518:f
514:h
487:f
456:2
452:V
448:L
443:D
440:g
435:f
431:h
427:2
421:=
418:f
401:f
370:0
354:,
336:0
313:0
297:,
294:0
288:{
283:=
277:y
269:u
246:y
242:u
214:0
30:.
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